On multichannel signal detection
- Ingster, Yuri I.
- Lepskii, Oleg V.
2010 Mathematics Subject Classification
- 62G10 62G20
- Multichannel signal detection, minimax hypothesis testing, adaptive hypothesis testing, distinguishability conditions
We consider 𝑛-channel signal detection system. Each 𝑖th channel could contain (or not contain) a signal. We suppose a signal is a function ƒ𝑖(t), t ∈ (0,1) observing in the white Gaussian noise of level ε > 0. Let 𝑘 be a number of channels which contain the signals. This number could be known or unknown. The functions ƒ𝑖 could be known or unknown as well. If shapes of functions ƒ𝑖 are unknown, then we consider nonparametric case. We suppose that functions ƒ𝑖 belongs to the Sobolev ball 𝚂σ where the smoothness parameter σ > 0 could be known or unknown as well. The cases, when 𝑘 or σ are unknown, lead to the "adaptive" problems. We are interested in the following problems:
(1) How large the signals ƒ𝑖 should be in order to detect these signals with vanishing errors, as the number of channels 𝑛 tends to infinity?
(2) What are the structures of test procedures which provide the detection of signals with the vanishing errors, if it is possible?
We show that there are two main types of results in the problems which, roughly, correspond to the cases either 𝑘 is "large" (this means 𝑘 >> 𝑛½ in the problem) or 𝑘 is "small" (this means 𝑘 << 𝑛½).